Population data: $2,5,8$

Part (a): Find the mean, $\mu$of the variable.

Part (b): For each of the possible sample sizes, construct a table similar to Table $7.2$on the page $293$and draw a dotplot for the sampling for the sampling distribution of the sample mean similar to Fig $7.1$on page $293$.

Part (c): Construct a graph similar to Fig $7.3$and interpret your results.

Part (d): For each of the possible sample sizes, find the probability that the sample mean will equal the population mean.

Part (e): For each of the possible sample sizes, find the probability that the sampling error made in estimating the population mean by the sample mean will be$0.5$or less, that is, that the absolute value of the difference between the sample mean and the population mean is at most $0.5$.

Consider the given question,

The population data is$2,5,8$.

The mean $\mu$is given below,

$$\begin{gathered} \mu =\frac{\sum _{x_i}}{N} \\ =\frac{2+5+8}{3} \\ =\frac{15}{3} \\ =5 \end{gathered}$$

For each of the possible sample sizes, we construct a table.

If the sample size taken $n=1$,

If the sample size taken $n=2$,

If the sample size taken $n=3$,

We can observe that from the dot plot there is one dot corresponding to $\mu =5$.

Hence, the probability that sample mean will be equal to population mean for $n=1$is role="math" localid="1652545895359" $\frac{1}{3}$.

Similarly, the probability that sample mean will be equal to population mean for data-custom-editor="chemistry" $\mathrm{n}=2$is $\frac{1}{3}$.

From the dot plot, we can see that there is no dot corresponding to $\mu =5$.

The probability that sample mean will be equal to population mean for$n=3$is$0$.

Number of dots within $0.5$or less of $\mu =5$is $1$out of $3$ when n is $1$and $2$.

For $n=3$, there is no dots are included in the sample mean $\overline{x}$will be $0.5$or less of $\mu$.

Hence, the probability that $\overline{x}$will be within $0.5$or less of $\mu$is for $n=1$is $\frac{1}{3}$.

Similarly, the probability that $\overline{x}$will be within $0.5$or less of $\mu$for $n=2$is $\frac{1}{3}$.

And the probability that $\overline{x}$ will be withindata-custom-editor="chemistry" $0.5$or less of$\mu$for$n=3$is$0$.